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CHAPTER - 2 Electromagnetism

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CHAPTER - 2 Electromagnetism Learning objectives Magnetic flux lines Force on current carrying conductor Biot Savart's law Magnetomotive force Permeability and relative permeability Reluctance Comparison of electric circuits and magnetic circuits Composite series magnetic circuits Leakage coefficient Electromagnetic induction Faraday's laws Lenz law Dynamically and statically induced emf Self and mutual inductance Coefficient of coupling Energy in a magnetic field 45 Uma-Rao: “chap-02” — 2008/1/1 — 00:41 — page 45 — #1 Electromagnetism 47 Coloumb first determined experimentally the quantitative expression for the magnetic force between two isolated poles. In reality magnetic poles cannot exist in isolation. Thus, the concept is purely theoretical. However, poles of a long thin magnet may be assumed to be isolated poles. The force between two magnetic poles placed in a medium is (i) directly proportional to their pole strengths m, (ii) inversely proportional to the square of the distance d between them (iii) inversely proportional to the absolute permeability of the medium. m m Km m F 1 2 or F 1 2 � ∝ µd2 = µd2 1 In SI system of units the value of K is 4π m1m2 m1m2 F 2 2 N (2.1) � = 4πµd = 4πµ0µr d where m1, m2 are the pole strengths, d is the distance in m, µ0 is permeability of free 7 space 4π 10 H/m, µr is relative permeability of the medium. = × − Thus, theoretically a unit magnetic pole may be defined as that pole which when placed in vaccum at a distance of one meter from a similar and equal pole repels it with a force of 1 N. Oersted discovered in 1820 that a magnetic field is produced around a current 4πµ0 carrying conductor. 2.2.1. Biot-Savart Law The expression for the magnetic field dB produced at a point P by an elemental length dl � � of a conductor carrying a current of I amperes is given by Biot-Savart’s law. Referring to Fig. 2.2. µIdl sin θ dB Wb/m2 � = 4πr2 or µIdl ar dB �×� Wb/m2 (2.2) � = 4πr2 Uma-Rao: “chap-02” — 2008/1/1 — 00:41 — page 47 — #3 48 Basic Electrical Engineering I dl ar θ P Figure 2.2 Biot-Savart’s law. where a is the unit vector along lines joining dl to P . The direction of dB is perpendicular r � to the plane� containing both dl and a . The field at a distance r due to an infinitely long � r straight conductor carrying a current�I amperes is given by µI B Wb/m2. � = 2πr The flux lines are in the form of concentric circles around the conductor. If the conductor is held with the thumb pointing in the direction of the current, the encircling fingers give the direction of the magnetic field. 2.2.2. Force on a current carrying conductor It was further observed that another current carrying conductor experiences a force when placed in the field. Now we can recollect that current is nothing but flow of electrons (charges!). Thus magnetic fields are produced by moving charges (current carrying con- ductor) and exert a force on moving charges. The characteristics of this magnetic force on a moving charge are as follows: Its magnitude is proportional to the magnitude of the charge. • The magnitude of the force is proportional to the magnitude or strength of the field. • The magnetic force depends on the particle’s (charge’s) velocity v. This is different • from the electric field force which is the same whether the charge is� moving or not. A charged particle at rest experiences no magnetic force. By experiment it is found that the force is always perpendicular to both the magnetic • field B and the velocity v. � � Uma-Rao: “chap-02” — 2008/1/1 — 00:41 — page 48 — #4 Electromagnetism 49 The above characteristics can be put compactly as, F qv B (2.3) � = � × � Similarly the force experienced by a current carrying conductor in a magnetic field is found to be proportional to the magnetic field B, the current I and the length of the � conductor and is perpendicular to the field and the length of the conductor. Thus, F Il B (2.4) � = �× � Since, the direction of the conductor, fixes the direction of the current (in space) (2.4) is more commonly written as F lI B (2.5) � = � × � Let F be the force in Newtons, I the current in amperes and l the length of the conductor � � in meters, at right amperes to the magnetic field. Then the magnetic field B or flux density � is the density of a magnetic field such that a conductor carrying a current of 1 ampere at right angles to the field has a force of 1 newton per meter acting upon it. The unit is Tesla (T), after the scientist Nikola Tesla. The force on a current carrying conductor is given by, F lIB sin θ (2.6) � = where θ is the angle between the magnetic field and the current carrying conductor. Thus a current carrying conductor experiences a force in the presence of a magnetic field. This principle is used in all electric motors. The direction of the force may be found from Fleming’s left-hand rule as shown in Fig. 2.3. force F magnetic field B current I left hand Figure 2.3 Uma-Rao: “chap-02” — 2008/1/1 — 00:41 — page 49 — #5 50 Basic Electrical Engineering Hold out your left hand with the fore finger, middle finger and thumb at right angles to each other. If the fore finger represents the direction of the field and the middle finger the direction of the current, the thumb gives the direction of the force on the conductor. From (2.5) it is obvious that no force is exerted on the conductor when it is parallel to the magnetic field (θ 0 ). = ◦ 2.2.3. Force between two current carrying conductors Consider two conductors carrying currents I1 and I2 respectively, separated by a distance of dm. The force between the conductors is attractive if the currents flow in the same direction and repulsive if the currents flow in opposite directions. Let us consider the force on the second current due to the first. The field produced by conductor 1 is given by µI B 1 T = 2πd The force experienced by conductor 2, from (2.5) is given by µlI I F 2 1 N = 2πd or the force per unit length is given by µI I F 1 2 N/m. = 2πd 2.2.4. Magnetic flux For a magnetic field having a cross-sectional area Am2 and a uniform flux density of B Teslas, the total flux in Webers (Wb) passing through a plane at right angles to the flow is given by φ BA = (Webers) (Tesla) (m2) = × or φ B (2.7) = A Uma-Rao: “chap-02” — 2008/1/1 — 00:41 — page 50 — #6 Electromagnetism 51 Hence the unit of B is also Wb/m2 1Tesla 1Wb/m2 = Example 2.1 A conductor carries a current of 500A at right angles to a magnetic field having a density of 0.4T. Calculate the force per unit length on the conductor. What would be the force if the conductor makes an angle of 45◦ to the magnetic field? Solution: F lI B lIB sin θ = � × � = When conductor is at right angles to the magnetic field, θ 90 . = ◦ F (1m)(500A)(0.4T) 200N/m. = = When θ 45 , = ◦ F (1m)(500A)(0.4T) sin 45◦ = × 141.42N/m. = Example 2.2 A rectangular coil 100mm by 150mm is mounted so that it rotates about the mid points of the 150mm sides. The axis of rotation is at right angles to a magnetic field with a flux density of 0.02T. Calculate the flux in the coil when (i) Maximum flux links with the coil. What is the position at which this occurs? (ii) The flux through the coil when the 150mm sides make an angle of 30◦ to the direction of flux. Solution: (i) This is shown in Fig. 2.4(a). The maximum flux passes through the coil when the plane of the coil is at right angles to the direction of the flux. 3 3 φ BA 0.02 (100 10− ) (150 10− ) = = × × × × 0.3mWb. = Uma-Rao: “chap-02” — 2008/1/1 — 00:41 — page 51 — #7 52 Basic Electrical Engineering axis of rotation 30° 0.02T 0.02T (a) (b) Figure 2.4 Example 2.1. (ii) This is shown in Fig. 2.4(b). 3 φ BAsin θ (0.3 10− ) sin 30◦ = = × × 0.15mWb. = 2.3. Magnetomotive force and magnetic field strength The magnetic flux is present in a magnetic circuit due to the existence of a magnetomotive force (mmf), caused by a current flowing through one or more turns. It is analogous to emf in an electric circuit which is responsible for the electric current. mmf NI (2.8) = where N is the number of turns. N is a dimensionless quantity. Hence, the unit of mmf is actually Ampere, though more commonly the unit is said to be ampere-turns (AT). Consider a coil as shown in Fig. 2.5. If the magnetic circuit is homogeneous and has a uniform cross sectional area, the mmf per metre length of the magnetic circuit is called the magnetic field strength H . NI H AT/m (2.9) = l The unit of H in SI units is A/m. The ratio B/H is the permeability, µ0, in free space B µ0 (2.10) = H Uma-Rao: “chap-02” — 2008/1/1 — 00:41 — page 52 — #8 Electromagnetism 53 N turns Figure 2.5 Coil with N turns on a toroid.
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